Schur's theorem

This article describes a result related to the sectional curvature of a Riemannian manifold

This result is valid in all dimensions

Statement

Let $(M,g)$ be a Riemannian manifold of dimension at least 3. Then the following are equivalent:

• For every point $p\in M$, the sectional curvature is constant across all tangent planes at $p$.
• The sectional curvature is constant (that is, the metric is a constant-curvature metric)

Proof

The idea behind the proof of this result is as follows. The second condition clearly implies the first, so what we essentially need to show is that the first implies the second.

For this, we convert the given condition on sectional curvature to a condition purely in terms of the Riemann curvature tensor.

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